{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Random Signals\n",
    "\n",
    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## White Noise"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Definition\n",
    "\n",
    "[White noise](https://en.wikipedia.org/wiki/White_noise) is a wide-sense stationary (WSS) random signal with constant power spectral density (PSD)\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{xx}(\\mathrm{e}^{\\,\\mathrm{j}\\, \\Omega}) = N_0\n",
    "\\end{equation}\n",
    "\n",
    "where $N_0$ denotes the power per frequency. White noise draws its name from the analogy to white light. It refers typically to an idealized model of a random signal, e.g. emerging from measurement noise. The auto-correlation function (ACF) of white noise can be derived by inverse discrete-time Fourier transformation (DTFT) of the PSD\n",
    "\n",
    "\\begin{equation}\n",
    "\\varphi_{xx}[\\kappa] = \\mathcal{F}_*^{-1} \\{ N_0 \\} = N_0 \\cdot \\delta[\\kappa]\n",
    "\\end{equation}\n",
    "\n",
    "This result implies that white noise has to be a zero-mean random process. It can be concluded from the ACF that two neighboring samples $k$ and $k+1$ are uncorrelated. Hence they show no dependencies in the statistical sense. Although this is often assumed, the probability density function (PDF) of white noise is not necessarily given by the normal distribution. In general, it is required to additionally state the amplitude distribution when denoting a signal as white noise."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example - Amplifier Noise\n",
    "\n",
    "Additive white Gaussian noise (AWGN) is often used as a model for amplifier noise. In order to evaluate if this holds for a typical audio amplifier, the noise captured from a microphone preamplifier at full amplification with open connectors is analyzed statistically. For the remainder, a function is defined to estimate and plot the PDF and ACF of a given random signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.stats as stats\n",
    "%matplotlib inline\n",
    "\n",
    "\n",
    "def estimate_plot_pdf_acf(x, nbins=50, acf_range=30):\n",
    "    \n",
    "    # compute and truncate ACF\n",
    "    acf = 1/len(x) * np.correlate(x, x, mode='full')\n",
    "    acf = acf[len(x)-acf_range-1:len(x)+acf_range-1]\n",
    "    kappa = np.arange(-acf_range, acf_range)\n",
    "    \n",
    "    # plot PSD\n",
    "    plt.figure(figsize = (10, 6))\n",
    "    plt.subplot(121)\n",
    "    plt.hist(x, nbins, density=True)\n",
    "    plt.title('Estimated PDF')\n",
    "    plt.xlabel(r'$\\theta$')\n",
    "    plt.ylabel(r'$\\hat{p}_x(\\theta)$')\n",
    "    plt.grid()\n",
    "\n",
    "    # plot ACF\n",
    "    plt.subplot(122)\n",
    "    plt.stem(kappa, acf, use_line_collection=True)\n",
    "    plt.title('Estimated ACF')\n",
    "    plt.ylabel(r'$\\hat{\\varphi}_{xx}[\\kappa]$')\n",
    "    plt.xlabel(r'$\\kappa$')\n",
    "    plt.axis([-acf_range, acf_range, 1.1*min(acf), 1.1*max(acf)]);\n",
    "    plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the pre-captured noise is loaded and analyzed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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   ],
   "source": [
    "noise = np.load('../data/amplifier_noise.npz')['noise']\n",
    "estimate_plot_pdf_acf(noise, nbins=100, acf_range=150)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inspecting the PDF reveals that it fits quite well to a [normal distribution](important_distributions.ipynb#Normal-Distribution). The ACF consists of a pronounced peak. from which can be concluded that the samples are approximately uncorrelated. Hence, the amplifier noise can be modeled reasonably well as additive white Gaussian noise. In oder to estimate the parameters of the normal distribution, the captured samples are fitted to a normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mean: -1.537e-05 \n",
      "Variance: 1.140e-07\n"
     ]
    }
   ],
   "source": [
    "mean, sigma = stats.norm.fit(noise)\n",
    "print('Mean: {0:1.3e} \\nVariance: {1:1.3e}'.format(mean, sigma**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Excercise**\n",
    "\n",
    "* What relative level does the amplifier noise have when the maximum amplitude of the amplifier is assumed to be $\\pm 1$?\n",
    "\n",
    "Solution: The average power of a mean-free random signal is given by is variance, here $\\sigma_\\text{noise}^2$. Due to the very low mean in comparison to the maximum amplitude, the noise can be assumed to be mean-free. Hence, the relative level of the noise is then given as $10*\\log_{10}\\left( \\frac{\\sigma_\\text{noise}^2}{1} \\right)$. Numerical evaluation yields"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Level of amplifier noise: -69.43 dB\n"
     ]
    }
   ],
   "source": [
    "print('Level of amplifier noise: {:2.2f} dB'.format(10*np.log10(sigma**2/1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example - Generation of White Noise with Different Amplitude Distributions\n",
    "\n",
    "Toolboxes for numerical mathematics like `Numpy` or `scipy.stats` provide functions to draw uncorrelated random samples with a given amplitude distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Uniformly distributed white noise**\n",
    "\n",
    "For samples drawn from a zero-mean random process with uniform amplitude distribution, the PDF and ACF are estimated as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
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      ],
      "text/plain": [
       "<Figure size 960x576 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "np.random.seed(3)\n",
    "estimate_plot_pdf_acf(np.random.uniform(size=10000)-1/2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets listen to uniformly distributed white noise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.io import wavfile\n",
    "fs = 44100\n",
    "\n",
    "x = np.random.uniform(size=5*fs)-1/2\n",
    "wavfile.write('uniform_white_noise.wav', fs, np.int16(x*32768))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<audio src=\"./uniform_white_noise.wav\" controls>Your browser does not support the audio element.</audio>[./uniform_white_noise.wav](./uniform_white_noise.wav)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Laplace distributed white noise**\n",
    "\n",
    "For samples drawn from a zero-mean random process with with Laplace amplitude distribution, the PDF and ACF are estimated as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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   "source": [
    "estimate_plot_pdf_acf(np.random.laplace(size=10000, loc=0, scale=1/np.sqrt(2)))"
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    "**Exercise**\n",
    "\n",
    "* Do both random processes represent white noise?\n",
    "* Estimate the power spectral density $N_0$ of both examples.\n",
    "* How does the ACF change if you lower the length `size` of the random signal. Why?\n",
    "\n",
    "Solution: Both processes represent white noise since the ACF can be approximated reasonably well as Dirac impulse $\\delta[\\kappa]$. The weight of the Dirac impulse is equal to $N_0$. In case of the uniformly distributed white noise $N_0 \\approx \\frac{1}{12}$, in case of the Laplace distributed white noise $N_0 \\approx 1$. Decreasing the length `size` of the signal increases the statistical uncertainties in the estimate of the ACF.  "
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   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples*."
   ]
  }
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